We tackle the problem of sampling from intractable high-dimensional density functions, a fundamental task that often appears in machine learning and statistics. We extend recent sampling-based approaches that leverage controlled stochastic processes to model approximate samples from these target densities. The main drawback of these approaches is that the training objective requires full trajectories to compute, resulting in sluggish credit assignment issues due to use of entire trajectories and a learning signal present only at the terminal time. In this work, we present Diffusion Generative Flow Samplers (DGFS), a sampling-based framework where the learning process can be tractably broken down into short partial trajectory segments, via parameterizing an additional "flow function". Our method takes inspiration from the theory developed for generative flow networks (GFlowNets), allowing us to make use of intermediate learning signals. Through various challenging experiments, we demonstrate that DGFS achieves more accurate estimates of the normalization constant than closely-related prior methods.

However, training and sampling from EBMs continue to pose significant challenges. The widely-used Denoising Score Matching (DSM) method [40] for scalable EBM training suffers from inconsistency issues, causing the energy model to learn a'noisy' data distribution. In this work, we propose an efficient sampling framework, (pseudo)-Gibbs sampling with moment matching, which enables effective sampling from the underlying clean model when given a'noisy' model that has been well-trained via DSM. We explore the benefits of our approach compared to related methods and demonstrate how to scale the method to high-dimensional datasets.

While generalized linear mixed models (GLMMs) are a fundamental tool in applied statistics, many specifications -- such as those involving categorical factors with many levels or interaction terms -- can be computationally challenging to estimate due to the need to compute or approximate high-dimensional integrals. Variational inference (VI) methods are a popular way to perform such computations, especially in the Bayesian context. However, naive VI methods can provide unreliable uncertainty quantification. We show that this is indeed the case in the GLMM context, proving that standard VI (i.e. mean-field) dramatically underestimates posterior uncertainty in high-dimensions. We then show how appropriately relaxing the mean-field assumption leads to VI methods whose uncertainty quantification does not deteriorate in high-dimensions, and whose total computational cost scales linearly with the number of parameters and observations. Our theoretical and numerical results focus on GLMMs with Gaussian or binomial likelihoods, and rely on connections to random graph theory to obtain sharp high-dimensional asymptotic analysis. We also provide generic results, which are of independent interest, relating the accuracy of variational inference to the convergence rate of the corresponding coordinate ascent variational inference (CAVI) algorithm for Gaussian targets. Our proposed partially-factorized VI (PF-VI) methodology for GLMMs is implemented in the R package vglmer, see https://github.com/mgoplerud/vglmer . Numerical results with simulated and real data examples illustrate the favourable computation cost versus accuracy trade-off of PF-VI.

The selection of the assumed effect size (AES) critically determines the duration of an experiment, and hence its accuracy and efficiency. Traditionally, experimenters determine AES based on domain knowledge. However, this method becomes impractical for online experimentation services managing numerous experiments, and a more automated approach is hence of great demand. We initiate the study of data-driven AES selection in for online experimentation services by introducing two solutions. The first employs a three-layer Gaussian Mixture Model considering the heteroskedasticity across experiments, and it seeks to estimate the true expected effect size among positive experiments. The second method, grounded in utility theory, aims to determine the optimal effect size by striking a balance between the experiment's cost and the precision of decision-making. Through comparisons with baseline methods using both simulated and real data, we showcase the superior performance of the proposed approaches.

Generative modeling is one of the most important tasks in machine learning, having numerous applications across vision (Rombach et al., 2022), language modeling (Brown et al., 2020), science (Ardizzone et al., 2018; Radev et al., 2020) and beyond. One of the best-motivated approaches to generative modeling is maximum likelihood training, due to its favorable statistical properties (Hastie et al., 2009). In the continuous setting, exact maximum likelihood training is most commonly achieved by normalizing flows (Rezende & Mohamed, 2015; Dinh et al., 2014; Kobyzev et al., 2020) which parameterize an exactly invertible function with a tractable change of variables (log-determinant term). This generally introduces a trade-off between model expressivity and computational cost, where the cheapest networks to train and sample from, such as coupling block architectures, require very specifically constructed functions which may limit expressivity (Draxler et al., 2022). In addition, normalizing flows preserve the dimensionality of the inputs, requiring a latent space of the same dimension as the data space.

This dissertation reports some first steps towards a compositional account of active inference and the Bayesian brain. Specifically, we use the tools of contemporary applied category theory to supply functorial semantics for approximate inference. To do so, we define on the `syntactic' side the new notion of Bayesian lens and show that Bayesian updating composes according to the compositional lens pattern. Using Bayesian lenses, and inspired by compositional game theory, we define fibrations of statistical games and classify various problems of statistical inference as corresponding sections: the chain rule of the relative entropy is formalized as a strict section, while maximum likelihood estimation and the free energy give lax sections. In the process, we introduce a new notion of `copy-composition'. On the `semantic' side, we present a new formalization of general open dynamical systems (particularly: deterministic, stochastic, and random; and discrete- and continuous-time) as certain coalgebras of polynomial functors, which we show collect into monoidal opindexed categories (or, alternatively, into algebras for multicategories of generalized polynomial functors). We use these opindexed categories to define monoidal bicategories of cilia: dynamical systems which control lenses, and which supply the target for our functorial semantics. Accordingly, we construct functors which explain the bidirectional compositional structure of predictive coding neural circuits under the free energy principle, thereby giving a formal mathematical underpinning to the bidirectionality observed in the cortex. Along the way, we explain how to compose rate-coded neural circuits using an algebra for a multicategory of linear circuit diagrams, showing subsequently that this is subsumed by lenses and polynomial functors.

The theory of belief functions [162, 67] is a modelling language for representing and combining elementary items of evidence, which do not necessarily come in the form of sharp statements, with the goal of maintaining a mathematical representation of an agent's beliefs about those aspects of the world which the agent is unable to predict with reasonable certainty. While arguably a more appropriate mathematical description of uncertainty than classical probability theory, for the reasons we have thoroughly explored in [50], the theory of evidence is relatively simple to understand and implement, and does not require one to abandon the notion of an event, as is the case, for instance, for Walley's imprecise probability theory [193]. It is grounded in the beautiful mathematics of random sets, and exhibits strong relationships with many other theories of uncertainty. As mathematical objects, belief functions have fascinating properties in terms of their geometry, algebra [207] and combinatorics. Despite initial concerns about the computational complexity of a naive implementation of the theory of evidence, evidential reasoning can actually be implemented on large sample spaces [156] and in situations involving the combination of numerous pieces of evidence [74]. Elementary items of evidence often induce simple belief functions, which can be combined very efficiently with complexity O(n + 1).

Noise is inevitably common in digital images, leading to visual image deterioration. Therefore, a suitable filtering method is required to lessen the noise while preserving the image features (edges, corners, etc.). This paper presents the efficient type-2 fuzzy weighted mean filter with an adaptive threshold to remove the SAP noise. The present filter has two primary steps: The first stage categorizes images as lightly, medium, and heavily corrupted based on an adaptive threshold by comparing the M-ALD of processed pixels with the upper and lower MF of the type-2 fuzzy identifier. The second stage eliminates corrupted pixels by computing the appropriate weight using GMF with the mean and variance of the uncorrupted pixels in the filter window. Simulation results vividly show that the obtained denoised images preserve image features, i.e., edges, corners, and other sharp structures, compared with different filtering methods.

We consider a combinatorial Gaussian process semi-bandit problem with time-varying arm availability. Each round, an agent is provided a set of available base arms and must select a subset of them to maximize the long-term cumulative reward. Assuming the expected rewards are sampled from a Gaussian process (GP) over the arm space, the agent can efficiently learn. We study the Bayesian setting and provide novel Bayesian regret bounds for three GP-based algorithms: GP-UCB, Bayes-GP-UCB and GP-TS. Our bounds extend previous results for GP-UCB and GP-TS to a combinatorial setting with varying arm availability and to the best of our knowledge, we provide the first Bayesian regret bound for Bayes-GP-UCB. Time-varying arm availability encompasses other widely considered bandit problems such as contextual bandits. We formulate the online energy-efficient navigation problem as a combinatorial and contextual bandit and provide a comprehensive experimental study on synthetic and real-world road networks with detailed simulations. The contextual GP model obtains lower regret and is less dependent on the informativeness of the prior compared to the non-contextual Bayesian inference model. In addition, Thompson sampling obtains lower regret than Bayes-UCB for both the contextual and non-contextual model.

For robotic decision-making under uncertainty, the balance between exploitation and exploration of available options must be carefully taken into account. In this study, we introduce a new variant of contextual multi-armed bandits called observation-augmented CMABs (OA-CMABs) wherein a decision-making agent can utilize extra outcome observations from an external information source. CMABs model the expected option outcomes as a function of context features and hidden parameters, which are inferred from previous option outcomes. In OA-CMABs, external observations are also a function of context features and thus provide additional evidence about the hidden parameters. Yet, if an external information source is error-prone, the resulting posterior updates can harm decision-making performance unless the presence of errors is considered. To this end, we propose a robust Bayesian inference process for OA-CMABs that is based on the concept of probabilistic data validation. Our approach handles complex mixture model parameter priors and hybrid observation likelihoods for semantic data sources, allowing us to develop validation algorithms based on recently develop probabilistic semantic data association techniques. Furthermore, to more effectively cope with the combined sources of uncertainty in OA-CMABs, we derive a new active inference algorithm for option selection based on expected free energy minimization. This generalizes previous work on active inference for bandit-based robotic decision-making by accounting for faulty observations and non-Gaussian inference. Our approaches are demonstrated on a simulated asynchronous search site selection problem for space exploration. The results show that even if incorrect observations are provided by external information sources, efficient decision-making and robust parameter inference are still achieved in a wide variety of experimental conditions.